# HEXAPOD – KINEMATICS

## Engineering Fundamentals

### INTRODUCTION

Hexapod or so called Stewart platform mechanisms are widely used in  precision engineering applications. The big advantage of this mechanism is the parallel linkage of all Degrees Of Freedom (DOF) from the moving platform to the base. In most cases this enables a much stiffer and compact design compared to a conventional mechanism where the independent DOF’s are stacked in a sequential way.

###### Geometry

The geometry of a generic hexapod system is depicted in the following figure. Index B for ‘Base’ and P for ‘Platform’

= Hinge locations in the base

= Radius of pitch circle of the hinge locations

= Angle between the hinge locations of a leg-pair

= Z-height of the platform in relation to the base

= Point of interest of the platform

###### Kinematics

The kinematics of a hexapod is typically derived via so called inverse kinematics. This means that actuator displacements are derived from platform displacements, and not vice versa. This is related to the fact that a given set of actuator displacements can theoretically correspond to up to 40 platform positions, whereas a given platform position corresponds to a single set of actuator displacements.

The derivation of the inverse kinematics is quite straight forward. First of all the coordinates of the nominal hinge locations are determined by

When assuming a platform displacement at point of interest

then the displacement of the platform hinge points can be captured by the following formula

With being the 3×3 unity matrix and Q (dRx,dRy,dRz) the rotation transformation matrix defined as

The displaced platform hinge locations are then

And the actuator displacement per leg is then calculated by

With this the inverse kinematics matrix can be derived as

such that the hexapod kinematics can be calculated by

Note that contains both translation as well as rotations, which typically imposes the use of dimensionless values referenced to SI units.

###### Range of motion

For hexapods the range of motion is difficult to capture as the range in 1 DOF is always dependent on the state of the other DOF’s. Looking to translations or rotations only, then the ranges can be captured by the following cube-like volume spanned by the corner points 18.
For a given geometry these corner points can be reconstructed numerically by the fact that the projection of points 27 on the XY / RxRy plane is an equilateral hexagon.